Skip to main content

Full text of "Treatise On Analysis Vol-Ii"

See other formats

100        XIII    INTEGRATION

This definition agrees with the preceding one when X is compact. It
expresses that the restriction of ju to Jf(X; K) is continuous with respect to
the topology induced by that of ^(X). We remark that tf (X; K) is closed
in ^c(X), and therefore a Banach space (3.14.5).

In general, a measure is not necessarily continuous on 3TC(X) with respect
to the topology induced by that of ^(X) (i.e., the topology defined by the
norm ||/||). We shall examine this question later (13.20).

Examples of Measures

(13.1.3)    Let X be a locally compact space, and let xeX. The mapping
/i->/(jc) of Jf (X) into C is a measure, for it is linear and we have \f(x)\ ^ ||/||
for each compact subset K of X such that/e Jf (X; K). This measure is called
the Dirac measure at the point x, or the measure defined by the unit mass at
the point jc, and is denoted by &x.

More generally, let (an) be a sequence of distinct points in X, and (/) a
sequence of complex numbers, such that for each compact K c X the sub-
sequence formed by the tnfor which an e K is absolutely summable (5.3). Let
CK  Z l*nl- Then, for each function /e ^T(X; K), the series ]T tnf(an) is

absolutely convergent, because the only nonzero terms are those for which
an e K, and we have

 \tJ(an)\ S II/H    E tn = cK-\\f\\.

an e K                                        an  K

This also shows that/i-> ^/fe) is a measure on X. This measure is said to


be defined by the masses tn at the points an for all n (cf. (13.18.8)).

(13.1.4)    Let/e JTCW- Fr each interval [a, b] containing the support of/,
the value of the integral     /(/) dt (8.7) is the same, and we denote it by

/+ oo                                                                       /*+ oo

f(f) dt. The mapping/r-^      /(/) dt is a linear form on JTCOR)> an(i it is

J ~~ oo                                                                                J  oo

a measure because, for each compact interval K = [a, b~] in R and each func-
tion/e jf(R; K) we have

+ 00


by the mean value theorem (8.7.7). This measure is called Lebesgue measure
on the real line R.

(13.1.5)    Let fj. be a measure on X and let g 6 #c(x)- Then for each function
/e X(X) it is clear that #/e jf (X), and the mapping/H-JU(#/) is therefore (13.14.4)).